Triangle Formulas
Shared 12/27/2025•213 views
Basic Definitions A triangle is a polygon with three edges and three vertices. Sum of interior angles: $A + B + C = 180^\circ$ (or $\pi$ radians). Types of Triangles Equilateral: All three sides are equal ($a=b=c$), all three angles are equal ($A=B=C=60^\circ$). Isosceles: Two sides are equal ($a=b$), the angles opposite those sides are equal ($A=B$). Scalene: All sides are different lengths, all angles are different. Right-angled: One angle is $90^\circ$. The side opposite the right angle is the hypotenuse. Acute: All three angles are less than $90^\circ$. Obtuse: One angle is greater than $90^\circ$. Area Formulas Base and Height: $Area = \frac{1}{2} \times base \times height = \frac{1}{2}bh$ Heron's Formula (given three sides $a, b, c$): $s = \frac{a+b+c}{2}$ (semi-perimeter) $Area = \sqrt{s(s-a)(s-b)(s-c)}$ Two Sides and Included Angle: $Area = \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$ Coordinates of Vertices $ (x_1, y_1), (x_2, y_2), (x_3, y_3) $: $Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ Perimeter $P = a+b+c$ Right-Angled Triangle Pythagorean Theorem: $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse). Trigonometric Ratios: $\sin \theta = \frac{opposite}{hypotenuse}$ $\cos \theta = \frac{adjacent}{hypotenuse}$ $\tan \theta = \frac{opposite}{adjacent}$ Laws for General Triangles Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (where $R$ is the circumradius) Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$ $a^2 = b^2 + c^2 - 2bc \cos A$ $b^2 = a^2 + c^2 - 2ac \cos B$ Special Points and Lines Centroid: Intersection of medians (lines from vertex to midpoint of opposite side). Divides median in $2:1$ ratio. Orthocenter: Intersection of altitudes (perpendiculars from vertex to opposite side). Circumcenter: Intersection of perpendicular bisectors of sides. Center of the circumcircle. Incenter: Intersection of angle bisectors. Center of the inscribed circle (incircle). Euler Line: Connects the orthocenter, centroid, and circumcenter (in non-equilateral triangles). Inradius and Circumradius Inradius ($r$): Radius of the incircle. $r = \frac{Area}{s}$ Circumradius ($R$): Radius of the circumcircle. $R = \frac{abc}{4 \times Area}$ Medians Length of median to side $a$ ($m_a$): $m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}$